En vue de l'obtention du

DOCTORAT DE L'UNIVERSITÉ DE TOULOUSEDélivré par :

Institut National Polytechnique de Toulouse (INP Toulouse)Discipline ou spécialité :

Réseaux, Télécommunications, Systèmes et Architecture

Présentée et soutenue par :Mme BOUCHRA BENAMMARle vendredi 5 décembre 2014

Titre :

Unité de recherche :

Ecole doctorale :

FORMES D'ONDES AVANCEES ET TRAITEMENTS ITERATIFS POURLES CANAUX NON LINEAIRES SATELLITES

Mathématiques, Informatique, Télécommunications de Toulouse (MITT)

Institut de Recherche en Informatique de Toulouse (I.R.I.T.)Directeur(s) de Thèse :

MME MARIE LAURE BOUCHERETMME NATHALIE THOMAS

Rapporteurs :M. CHRISTOPHE LAOT, TELECOM BRETAGNE CAMPUS DE BREST

M. PIERRE DUHAMEL, SUPELEC

Membre(s) du jury :1 M. PHILIPPE CIBLAT, TELECOM PARISTECH, Président2 M. CHARLY POULLIAT, INP TOULOUSE, Membre2 M. MARCO LOPS, UNIVERSITA DEGLI STUDI DI CASSINO, Membre2 M. MATHIEU DERVIN, THALES ALENIA SPACE, Membre2 Mme MARIE LAURE BOUCHERET, INP TOULOUSE, Membre2 Mme NATHALIE THOMAS, INP TOULOUSE, Membre

ii

Remerciements

Je tiens tout d'abord à remercier mes encadrants de thèse pour m'avoir fait con�ance, et soutenu du-

rant ma thèse. Plus spécialement, je remercie Marie-Laure Boucheret pour ses éclairages salutaires au

sujet des modulations en mono et multi-porteuse, et que personne ne s'avise de dire que le SC-FDMA

est une modulation mutli-porteuse au risque de s'attirer les foudres de Marie-Laure. Je remercie aussi

Nathalie Thomas, pour ses nombreuses relectures et pour sa gentillesse en toute circonstance qui ont

eu un e�et ô combien réconfortant (merci pour les bonbons). Mes remerciements vont aussi à Charly

Poulliat pour m'avoir guidé tout au long de ma thèse et avoir été présent à toute heure a�n de me

relire, qu'il trouve en mes mots toute ma gratitude.

Durant ces trois années de thèse, j'ai aussi eu des échanges avec Mathieu Dervin ingénieur de

recherche à Thalès. Je souhaiterai le remercier pour sa disponibilité. Il a sû répondre de manière très

pédagogique à mes innombrables questions aux sujets des enjeux des systèmes actuels.

Je remercie aussi Pierre Duhamel Directeur de recherche CNRS au sein du Laboratoire Signaux

et Systèmes de Paris ainsi que Christophe Laot Professeur à TELECOM Bretagne qui m'ont fait

l'honneur de rapporter ma thèse. Que soient aussi remerciés Philippe Ciblat professeur à TELECOM

ParisTech et Marco Lops professeur à l'université de Cassino en Italie pour avoir accepté d'examiner

ma thèse.

Ensuite, parce que la vie du laboratoire n'aurait pas été aussi agréable sans eux, je souhaite

remercier de tout coeur tous mes amis et collègues de laboratoire. Plus spécialement je remercie,

Abdé, Cécilé, Yoann, Qi, Ningning, Sébastien, Olivier (dit Jack-Sparrow), Sokchenda (agent SS),

Romain, Bilel, Farouk, Ahmad, Mohammad, Mohamed, Aziz, Nesrine, JB (the hat man), Farouk,

iii

Facundo, Emilie, Jean-Gabriel (dit le corse) et Hermine. N'oublions pas les TeSA-people (les gens de

l'au-delà ... du canal) Raoul, Victor, Jorge (dit Georges), Jean-Adrien (l'acteur), Tarik (le penseur)

et Fabio. Comment oublier ces parties de Hanabi, de contact, de bowling ou de speed-cubing ? Que

cela soit sur le terrain de foot, perchés au sommet du Montségur, en pique-nique à Pech David ou

mieux encore, a�alés sur la piste de la patinoire, vous avez su rendre inoubliable mon séjour parmi

vous. Je vous souhaite à tous tout le bonheur et la réussite que l'on peut espérer.

Mes derniers remerciements vont à ceux sans qui rien de tout cela n'aurait été possible. Je remercie

très chaleureusement mes parents pour leur sacri�ces a�n de me permettre de venir étudier en France,

mon frère et en�n ma moitié pour m'avoir soutenue durant ma thèse et avoir partagé mes joies et

mes peines. Je souhaite que vous trouviez dans mes mots l'expression de mon sincère respect et amour.

Bouchra

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Résumé

L'augmentation de l'e�cacité spectrale des transmissions mono-porteuses sur un lien de di�usion par

satellite est devenu un dé� d'envergure a�n de pallier la demande croissante en débits de transmission.

Si des techniques émergentes de transmissions encouragent l'utilisation de modulations à ordre élevé

telles que les modulations de phase et d'amplitude (APSK), certaines dégradations sont encourues lors

du traitement à bord du satellite. En e�et, en raison de l'utilisation d'ampli�cateurs de puissance ainsi

que de �ltres à mémoires, les modulations d'ordre élevé subissent des distorsions non-linéaires dues à la

�uctuation de leur enveloppe, ce qui nécessite des traitements au sein de l'émetteur ou bien au sein du

récepteur. Dans cette thèse, nous nous intéressons au traitement de l'interférence non-linéaire au sein

du récepteur, avec une attention particulière aux égaliseurs itératifs qui améliorent les performances du

système au prix d'une complexité élevée. A partir du modèle temporel des interférences non-linéaires

induites par l'ampli�cateur de puissance, des algorithmes de réception optimaux et sous optimaux

sont dérivés, et leurs performances comparées. Des égaliseurs à complexité réduite sont aussi étudiés

dans le but d'atteindre un compromis performances-complexité satisfaisant. Ensuite, un modèle des

non-linéarités est dérivé dans le domaine fréquentiel, et les égaliseurs correspondants sont présentés.

Dans un second temps, nous analysons et dérivons des récepteurs itératifs pour l'interférence entre

symboles non linéaire. L'objectif est d'optimiser les polynômes de distributions d'un code externe

basé sur les codes de contrôle de parité à faible densité (LDPC) a�n de coller au mieux à la sortie

de l'égaliseur. Le récepteur ainsi optimisé atteint de meilleures performances comparé à un récepteur

non optimisé pour le canal non-linéaire. Finalement, nous nous intéressons à une classe spéci�que

de techniques de transmissions mono-porteuse basée sur le multiplexage par division de fréquence

v

(SC-OFDM) pour les liens satellites. L'avantage de ces formes d'ondes réside dans l'e�cacité de leur

égaliseur dans le domaine fréquentiel. Des formules analytiques de la densité spectrale de puissance

et du rapport signal sur bruit et interférence sont dérivées et utilisées a�n de prédire les performances

du système.

Publications

Journals in preparation

1. B.Benammar, N.Thomas, C.Poulliat, ML.Boucheret, M.Dervin, "Iterative Receivers For Non Linear

Satellite Channels" to be submitted to IEEE trans. on Communications.

2. B.Benammar, N.Thomas, C.Poulliat, ML.Boucheret, M.Dervin, "Performance Analysis Of Block Circu-

lar Filter-Bank Modulations" to be submitted.

International conferences

Accepted

1. B.Benammar, N.Thomas, C.Poulliat, ML.Boucheret, M.Dervin, "Asymptotic Analysis and Design of

Iterative Receivers for Non Linear ISI Channels" ISTC August 18-22, 2014, Bremen, Germany.

2. B.Benammar, N.Thomas, C.Poulliat, ML.Boucheret, M.Dervin, "On Linear Frequency Domain Turbo-

Equalization of Non Linear Volterra Channels" ISTC August 18-22, 2014, Bremen, Germany.

3. H.Abdulkader, B.Benammar, C.Poulliat, ML.Boucheret, N.Thomas, "Analysis and Design of Radial

Basis Function-Based Turbo Equalizers" ISTC August 18-22, 2014, Bremen, Germany.

4. H.Abdulkader, B.Benammar, C.Poulliat, ML.Boucheret, N.Thomas, "Neural Networks-Based Turbo

Equalization of a Satellite Communication Channel" SPAWC June 22-25, 2014, Toronto, Canada.

5. B.Benammar, N.Thomas, C.Poulliat, ML.Boucheret, M.Dervin, "On linear MMSE Based Turbo-equalization

of Nonlinear Satellite Channels" ICASSP May 26-31, 2013, Vancouver, Canada.

6. B.Benammar, N.Thomas, ML.Boucheret, C.Poulliat, M.Dervin, " Analytical expressions of Power Spec-

tral Density for General Spectrally Shaped SC-FDMA Systems" EUSIPCO Sep. 9-13, 2013, Marrakech,

Morocco.

vi

National conferences

1. B.Benammar, N.Thomas, C.Poulliat, ML.Boucheret, M.Dervin, "Turbo- Egalisation MMSE Lineaire De

Canaux Non Lineaires" GRETSI Sep. 3-6, 2013, Bretagne, France.

vii

viii

Abstract

Increasing both the data rate and power e�ciency of single carrier transmissions over broadcast satellite links

has become a challenging issue to comply with the urging demand of higher transmission rates. If emerging

transmission techniques encourage the use of high order modulations such as Amplitude and Phase Shift

Keying (APSK) and Quadrature Amplitude Modulation (QAM), some channel impairments arise due to on-

board satellite processing. Indeed, due to satellite transponder Power Ampli�ers (PA) as well as transmission

�lters, high order modulations incur non linear distortions due to their high envelope �uctuations which require

speci�c processing either at the transmitter or at the receiver.

In this thesis, we investigate on non linear interference mitigation at the receiver with a special focus on iterative

equalizers which dramatically enhance the performance at the cost of additional complexity. Based on the time

domain model of the non linear interference induced by the PA, optimal and sub-optimal receiving algorithms

are proposed and their performance compared. Low complexity implementations are also investigated for the

sake of a better complexity-performance trade-o�. Then, a non linear frequency domain model is derived and

the corresponding frequency equalizers are investigated.

In the second part, we analyse and design an iterative receiver for the non linear Inter Symbol Interference

(ISI) channel. The objective is to optimize an outer Low Density Parity Check (LDPC) code distribution

polynomials so as to best �t the inner equalizer Extrinsic information. The optimized receiver is shown to

achieve better performance compared to a code only optimized for linear ISI channel.

Finally, we investigate on a speci�c class of single carrier transmissions relying on Single Carrier Orthogonal

Frequency Division Multiplexing (SCO-FDM) for satellite downlink. The advantage of such waveforms lies

in their practical receiver implementation in the frequency domain. General analytical formulas of the power

spectral density and signal to noise and interference ratio are derived and used to predict the bit error rate for

frequency selective multiplexers.

ix

x

Abreviations

AM-AM Amplitude to Amplitude

AM-PM Amplitude to Phase

BCH Bose, Ray-Chaudhuri et Hocquenghem

BEC Binary Erasure Channel

BICM Bit Interleaved Coded Modulation

BP Belief Propagation

BPSK Binary Phase Shift Keying

CA Complex Adds

CC Convolutional Codes

CCDF Cumulative Complementary Density Function

CM Complex Multiplies

CN Check Node

DFT Discrete Fourier Transform

DVB-S Digital Video Broadcasting Satellite

ENGINES Enabling Next GeneratIon NEtworks for broadcast Services

ETSI European Telecommunications Standards Institute

EW-SC-FDM Extended Weighted Singe-Carrier Frequency Division Multiplexing

EXIT EXtrinsic Information Transfer

FET Field E�ect Transistor

xi

FIR Finite Impulse Response

FSS Fixed Services Satellite

GF Galois Field

GFDM Generalised Frequency Division Multiplexing

GM Gaussian Mixture

IBO Input Back-O�

IMUX Input MUltipleXer

LDPC Low Density Parity Check

MAP Maximum A Posteriori

MLSE Maximum Likelihood Sequence Equalizer

MMSE Minimum Mean Square Error

MSE Mean Square Error

MSS Mobile Services Satellite

OFDM Orthogonal Frequency Division Multiplexing.

OFDMA Orthogonal Frequency Division Multiple Access.

OMUX Output MUltipleXer

OBO Output Back-O�

PAPR Peak to Average Power Ratio

QAM Quadrature and Amplitude modulations

QPSK Quadrature Phase Shift Keying

RF Radio Frequency

RS Reed Solomon

SISO Soft Input Soft Output

SP Sum Product (SP)

SSPA Solid State Power Ampli�er

TT & C Telemetric Tracking and Command

TWTA Travelling Wave Tube Ampli�er

UE User Equipment

VN Variable Nodexii

Contents

Remerciements iii

Résumé v

Abstract ix

Abreviations xi

Introduction (French) 1

1 Digital communications over satellite channels 9

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Introduction (french) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Satellite communication system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Broadcasting Satellites standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 DVB-S standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.2 DVB-S2 standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.3 DVB-S2X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Transponder modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5.1 TWT and SSP Ampli�ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5.2 Input and output multiplexers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5.3 Saturation levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6 Impact of system parameters on the non linear channel . . . . . . . . . . . . . . . . . . . . . . . 22

1.6.1 Impact of IBO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6.2 Impact of the root raised cosine roll-o� . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

xiii

1.6.3 Impact of the signal bandwidth in the presence of IMUX and OMUX . . . . . . . . . . 24

1.7 Non linear satellite channel models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.7.1 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.7.2 Volterra model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.7.3 Volterra coe�cients design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.7.4 Volterra decomposition for test channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.8 Frequency domain Volterra model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.8.1 Multi-dimensional Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.8.2 Frequency domain Volterra model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.10 Conclusion (french) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2 Mitigation of non linear satellite channels interference 41

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2 Introduction (French) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 Optimal time domain equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.1 Trellis based structure of the non linear channel . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.2 Symbol based detection: MAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.3 Sequence based detection: MLSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4 Linear time domain equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.5 Non linear sub-optimal equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5.1 Non linear adaptive Volterra equalization . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5.2 Decision Feedback Equalization: DFE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.6 Frequency domain equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.6.1 Linear MMSE -FD equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.6.2 Hybrid time and frequency domain DFE . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.7 Equalization schemes comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.7.1 Complexity comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.7.2 Performance comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.9 Conclusion (french) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

xiv

3 Iterative equalization and decoding 69

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2 Introduction (French) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.3 Turbo equalization principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.4 Optimal SISO MAP equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.5 Linear MMSE turbo-equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.5.1 Linear MMSE time varying solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.5.2 No-Apriori (NA) MMSE approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.5.3 Averaged Low Complexity (ALC) MMSE approximation . . . . . . . . . . . . . . . . . . 82

3.5.4 Frequency domain turbo linear MMSE equalizer . . . . . . . . . . . . . . . . . . . . . . 82

3.6 SISO MAP decoding over a trellis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.7 SISO LDPC decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.7.1 Useful notations and de�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.7.2 Belief Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.8 Comparison of iterative equalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.8.1 Complexity comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.8.2 Performance comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.9 Receiver asymptotic analysis and design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.9.1 Mutual information computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.10 Asymptotic code design using EXIT charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.10.1 Iterative receiver scheduling and interleaver assumptions . . . . . . . . . . . . . . . . . . 101

3.10.2 Code optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.10.3 Optimization results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.12 Conclusions (French) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4 SC-OFDM in satellite communications 109

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2 Introduction (French) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.3 SC-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.3.1 Frequency based SC-OFDM scheme description . . . . . . . . . . . . . . . . . . . . . . . 113

4.3.2 Extended Weighted SC-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

xv

4.4 From frequency to time domain representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.4.1 Multi-rate FFT/IFFT noble identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.4.2 Transmitter (Tx) equivalent model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.4.3 Receiver (Rx) modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.4.4 Global system time domain equivalent model . . . . . . . . . . . . . . . . . . . . . . . . 119

4.4.5 EW-SC-OFDM as a circular convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.4.6 Special cases of the general scheme: SC-OFDM and EW-SC-OFDM . . . . . . . . . . . 122

4.5 PSD analysis of SC-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.5.1 PSD with rectangular shaping : SC-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.5.2 PSD with root raised cosine EW-SC-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.6 Linear equalization and SINR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.6.1 The useful term power Pu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.6.2 The interfering term power σ2i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.6.3 The noise power σ2w̃ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.6.4 SINR function of SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.6.5 Linear equalizers: MMSE and ZF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.7 Applications to the SINR of SC-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.7.1 SINR of SC-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.7.2 SINR of EW-SC-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.9 Conclusion (French) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5 Conclusions and future work 135

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2.1 Estimation canal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2.2 Synchronisation horloge et porteuse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2.3 Bourrage de zéros ou bien cyclique pré�xe ? . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2.4 Comparaison avec d'autres techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2.5 Égalisation au sens large . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.2.6 Generalised Frequency division multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

xvi

5.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.4.1 Channel estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.4.2 Frequency and clock synchronisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.4.3 Zero padding or cyclic pre�xing? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.4.4 Comparison with alternative techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.4.5 Widely linear equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.4.6 Generalised Frequency division multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . 143

Appendices 147

A Analytical expressions of the power spectral density of SC-OFDM 147

B Equivalent noise variance 151

C On How to estimate the ellipse parameters of a non-circular noise 153

C.0.7 Ellipse characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

C.0.8 Application to a correlated noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

xvii

xviii

List of Figures

1.1 DVB-S functional block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 DVB-S2 functional block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Scatter plot of the DVB-S2 modulation γ = 2.85 and γ1 = 2.84 and γ2 = 5.27 . . . . . . . . . . 14

1.4 Structure of a Travelling Wave Tube Ampli�er . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 AM-AM characteristic using Saleh's model with parameters αa = 1.9638 and βa = 0.9945 . . . 18

1.6 AM-PM characteristic using Saleh's model with parameters αφ = 2.5293 and βφ = 2.8168 . . . 19

1.7 Gain for IMUX and OMUX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.8 Group delay for IMUX and OMUX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.9 Satellite channel modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.10 Constellation for 16APSK with roll-o� α = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.11 PDF of real and imaginary parts of one outer ring centroid at IBO = 0dB . . . . . . . . . . . . 24

1.12 PDF of real and imaginary part of one outer ring centroid at IBO = 9dB . . . . . . . . . . . . 25

1.13 Power spectral density of the ampli�ed signal for di�erent IBO values . . . . . . . . . . . . . . 26

1.14 ISI variance function of the root raised cosine roll-o� α . . . . . . . . . . . . . . . . . . . . . . . 27

1.15 Constellation warping for 16APSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.16 Polynomial decomposition for AM-AM Saleh's function . . . . . . . . . . . . . . . . . . . . . . 32

1.17 Polynomial decomposition for AM-PM Saleh's function . . . . . . . . . . . . . . . . . . . . . . 33

1.18 Scatterplots of 16APSK with roll-o� α = 0.2 at IBO = 3dB . . . . . . . . . . . . . . . . . . . . 34

1.19 System model description in the frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1 Trellis representation of the non linear channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2 DFE with linear feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3 DFE with non linear feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4 Hybrid Time and frequency time domain equalizer . . . . . . . . . . . . . . . . . . . . . . . . . 61

xix

2.5 Mean Square Error for the MMSE time domain equalizer function of the feed-forward �lter

lengths N1 and N2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.6 Number of complex MMSE operations for a block of estimated symbols L = 512 . . . . . . . . 66

2.7 Bit Error Rate performance of the equalizers over the channel test 2 . . . . . . . . . . . . . . . 67

3.1 Structure of a turbo equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2 Iterative Volterra receiver for the non linear channel . . . . . . . . . . . . . . . . . . . . . . . . 73

3.3 Linear MMSE time domain solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.4 Estimation error PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.5 Common sub-matrix in the MMSE solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.6 SIC MMSE turbo FDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.7 Tanner graph representation of the LDPC code matrix H where (N = 8, dv = 2, dc = 4) . . . . 88

3.8 Belief propagation for a degree i VN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.9 Belief propagation for a degree j CN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.10 Complexity comparison for iterative receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.11 Performance comparison of the non linear channel iterative receivers . . . . . . . . . . . . . . . 93

3.12 Gaussian mixture LLRs for 16APSK rate 3/4 at Eb/N0 = 20dB . . . . . . . . . . . . . . . . . . 97

3.13 EXIT charts for the AWGN 16APSK BICM soft demapper . . . . . . . . . . . . . . . . . . . . 98

3.14 Mutual Information for the 16APSK-BICM soft demapper using di�erent approximations . . . 100

3.15 The function Ψ(σ) and its inverse Ψ−1(Ie) for 16-APSK BICM . . . . . . . . . . . . . . . . . . 101

3.16 Mutual Information for the 16APSK-BICM MAP and MMSE equalizers for test channel 2 . . . 102

3.17 Global scheme of a satellite communication channel. GM stands for quantities with a Gaussian

Mixture approximation, G for Gaussian approximation . . . . . . . . . . . . . . . . . . . . . . . 102

3.18 Partial and ensemble interleaving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.19 Achievable designed rates compared with the MAP optimal equalizer and the 16-APSK ISI-free

rates for a maximum dv = 10 over the test channel 2 . . . . . . . . . . . . . . . . . . . . . . . . 105

3.20 Achievable designed rates compared with the NA-MMSE equalizer and the 16-APSK ISI-free

rates for a maximum dv = 10 over the test channel 2 . . . . . . . . . . . . . . . . . . . . . . . . 105

3.21 Bit error rate for the iterative receivers and the optimized LDPC code . . . . . . . . . . . . . . 106

4.1 IMUX/OMUX responses for di�erent symbol rates . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.2 SC-FDMA and SC-OFDM frequency based representation . . . . . . . . . . . . . . . . . . . . . 113

4.3 EW-SC-OFDM transmission scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.4 Exteding from a length M to U ≤ 2M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

xx

4.5 CCDF of INP for SC-OFDM, EW-SC-OFDM (roll-o�s 0.05 and 0.25) and OFDM usingN = 512

and M = 432 with an oversampling factor 4 for 16APSK . . . . . . . . . . . . . . . . . . . . . . 116

4.6 Up-sampling identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.7 Down-sampling identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.8 An example of stacking with L = 8, N = 4, and LN = 2 . . . . . . . . . . . . . . . . . . . . . . 117

4.9 Localised mapping modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.10 Transmitter equivalent model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.11 Localised demapping and equalization equivalent model . . . . . . . . . . . . . . . . . . . . . . 119

4.12 Receiver equivalent model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.13 SC-OFDM equivalent frequency domain system model . . . . . . . . . . . . . . . . . . . . . . . 120

4.14 SC-OFDM equivalent time domain system model . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.15 Receiver structure of SC-OFDM using spectral shaping of length U ≥M . . . . . . . . . . . . . 1214.16 Spectral shaping �lter with root raised cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.17 Transmitter model of SC-OFDM with pulse shaping . . . . . . . . . . . . . . . . . . . . . . . . 123

4.18 PSD EW-SC-OFDM with M = 426, N = 512, and α = 0 . . . . . . . . . . . . . . . . . . . . . 124

4.19 PSD EW-SC-OFDM with M = 426, N = 512, and α = 0.25 . . . . . . . . . . . . . . . . . . . . 125

4.20 SC-OFDM simpli�ed system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.21 Equivalent noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.22 MMSE and ZF SINR for EW-SC-OFDM N = 512, M = 426, and α = 0.05 . . . . . . . . . . . 132

4.23 MMSE and ZF BER for EW-SC-OFDM N = 512, M = 426, and α = 0.05 . . . . . . . . . . . . 133

5.1 EW-SC-OFDM with roll-o� α = 0.05, N = 512 and M = 426 . . . . . . . . . . . . . . . . . . . 138

5.2 Time domain GFDM system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

C.1 Approximations to the .99th quantile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

C.2 Approximations to the .90th quantile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

xxi

xxii

List of Tables

1.1 DVB-S2 modulation schemes parameters for QPSK, 8PSK, 16APSK and 32ASPK . . . . . . . 15

1.2 Optimum constellation radius ratio for AWGN channel with 16APSK . . . . . . . . . . . . . . 15

1.3 Optimum constellation radius ratio for AWGN channel with 32APSK . . . . . . . . . . . . . . 16

1.4 Comparative characteristics for TWTA and SSPA . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 ISI variance for di�erent signal bandwidths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.6 Table of MSE for di�erent polynomial order decompositions . . . . . . . . . . . . . . . . . . . . 32

1.7 Test channels characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.8 Volterra kernels for test channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1 Comparison of linear and non linear Volterra channel equalizers . . . . . . . . . . . . . . . . . . 64

3.1 Approximation parameters for the J and J−1 function . . . . . . . . . . . . . . . . . . . . . . . 95

3.2 Gaussian Mixture parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.1 Values for the over-all channel number of symbol spaced taps . . . . . . . . . . . . . . . . . . . 112

4.2 Simulation parameters for SC-OFDM and EW-SC-OFDM . . . . . . . . . . . . . . . . . . . . . 131

xxiii

xxiv

Introduction (French)

Pendant plus d'un demi siècle, les systèmes satellites ont acheminé et transporté des données vers des con-

trées lointaines et sur des zones étendues. Que ce soit pour les télécommunications, le positionnement ou

l'observation de la terre, l'augmentation des débits des communications par satellite revêt une importance

majeure dans les évolutions futures de ces technologies.

Lorsqu'il s'agit de développer un système de communication, plusieurs paramètres entrent en jeu a�n de di-

mensionner les capacités des liens. Plus précisément, en fonction de la nature du service rendu par le satellite

(�xe, mobile, di�usion), du sens de la communication (liaison montante ou descendante), de la bande passante

disponible ainsi que de la complexité permise, les solutions permettant d'augmenter le débit et l'e�cacité en

puissance peuvent di�érer.

L'e�cacité en puissance mesure la robustesse face aux perturbations du bruit et est souvent liée à la distance

minimale du code et de la modulation utilisés. L'e�cacité spectrale quant à elle, caractérise le débit de la

communication pour une bande donnée et est reliée à la cardinalité de la modulation ainsi qu'au rendement de

codage. Il existe en général un compromis entre e�cacité en puissance et e�cacité spectrale. Cependant, de

nouvelles avancées dans le domaine des traitements itératifs ont permis de réduire ce compromis en exploitant

un degré de liberté supplémentaire qui est celui de la complexité.

Dans cette thèse, nous nous positionnons au niveau de la voie descendante d'un système de di�usion par satel-

lite. Pour une telle application, l'élément dimensionnant est le transpondeur à bord du satellite qui comprend

les �ltres multiplexeurs ainsi que l'ampli�cateur de puissance. En e�et, puisque le satellite est contraint en

termes de consommation de puissance et d'interférence sur les canaux adjacents, le transpondeur est ainsi "le

goulot" de l'optimisation du système. Nous étudions donc deux techniques permettant d'augmenter les débits

du système et évaluons leur impact sur les éléments du transpondeur.

D'une part, l'utilisation de modulations d'ordre élevé permet d'augmenter le nombre de bits transmis par utili-

sation du canal, mais entraine pour les modulations d'amplitude et de phase une augmentation de la �uctuation

1

2 Introduction

de l'enveloppe du signal. Ceci donne lieu à de l'interférence non linéaire quand ces signaux sont traités par un

ampli�cateur de puissance opérant à saturation (ou proche de la saturation). Ce type d'interférence est aussi

rencontré dans d'autres applications telles que les canaux à enregistrement magnétique ou les communications

par �bre optique. D'autre part, augmenter les débits utiles peut aussi être réalisé, en augmentant le débit du

signal. Cependant, si la bande du signal dépasse celle des �ltres multiplexeurs, la sélectivité en fréquence du

canal satellite est augmentée, générant une interférence additionnelle à la réception.

Pour les deux techniques d'augmentation de l'e�cacité spectrale, une analyse de la nature et de la quantité

d'interférence est nécessaire a�n d'adopter les méthodes de réduction d'interférences adéquates. Les traite-

ments peuvent ainsi être envisagés au niveau de l'émetteur au travers de techniques dites de pré-compensation

ou de pré-distorsion, ou au niveau du récepteur par des techniques dites d'égalisation. Dans cette thèse nous

nous intéressons plus particulièrement à l'analyse et l'optimisation des récepteurs itératifs pour le canal non

linéaire, ainsi qu'aux formes d'ondes nouvelles permettant des algorithmes de réception plus e�caces.

Structure du manuscrit

Cette thèse s'articule en quatre chapitres principaux dont voici les descriptifs:

• Chapitre 1: Ce chapitre présente une description du système satellite surle quel notre étude portera.Il s'agit d'un système de di�usion par satellite et plus spéci�quement du standard de di�usion vidéo

numérique (DVB). Nous présentons les nouvelles modulations proposées dans le standard DVB-S2 qui

permettent d'atteindre un compromis intéressant entre e�cacité en puissance et e�cacité spectrale. Nous

étudions ensuite les éléments constituants du transpondeur à bord du satellite, en évaluant l'impact de

paramètres tels que le roll-o� des �ltres, la bande du signal, les reculs de l'ampli�cateur sur la quantité

et la forme de l'interférence reçue. Ensuite, nous présentons un modèle de l'interférence non linéaire

au rythme symbole en nous basant sur une décomposition en somme in�nie dite de Volterra. Pour des

raisons de complexité, ce modèle est tronqué aux raisonnables troisième et cinquième ordres, et l'impact

de cette troncature sur la précision du modèle est évalué. En�n, le modèle fréquentiel équivalent de la dé-

composition en séries de Volterra est décrit et sera utilisé ultérieurement pour les traitements fréquentiels.

• Chapitre 2: Ce chapitre traite des égaliseurs non itératifs pour le modèle non linéaire du canalsatellite. Dans un premier temps, nous présentons une description sous forme de chaîne de Markov des

non linéarités qui permet de dériver des égaliseurs optimaux au sens symbole et séquence. En raison

3

de la complexité exponentielle de ces égaliseurs, nous étudions des égaliseurs sous optimaux linéaires et

non linéaires. Plus précisément, nous développons les expressions d'égaliseurs linéaires qui minimisent

l'erreur quadratique moyenne ainsi que deux égaliseurs non linéaires à retours de décision. Ensuite,

d'une manière similaire au domaine temporel, nous présentons des expressions d'égaliseurs linéaires et

non linéaires fréquentiels. Nous dérivons ainsi de nouveaux résultats concernant l'égalisation hybride

temps-fréquence pour le canal de Volterra. En�n, ces di�érentes implémentations sont comparées en

termes de taux d'erreurs binaires et de complexité.

• Chapitre 3: Ce chapitre présente des résultats sur la turbo égalisation du canal non linéaire satellite.Premièrement, nous rappelons des résultats sur l'égalisation itérative optimale pour des canaux décrits

par un treillis, ce qui est le cas du canal nonlinéaire modélisé par une série de Volterra. Ensuite, nous

dérivons des expressions pour les égaliseurs linéaires basés sur le modèle de Volterra, et étudions les

approximations à faible complexité de ces égaliseurs. Par ailleurs, nous analysons l'égalisation itérative

fréquentielle du canal de Volterra. Dans un second temps, nous concevons et optimisons le code canal qui

permet de s'adapter au mieux aux messages issus de l'égaliseur en utilisant la méthode d'ajustement de

la courbe (curve �tting) en utilisant l'outil EXIT. Pour ce faire, nous modélisons la sortie de l'égaliseur

par un mélange de Gaussiennes qui est plus adéquat que l'approximation Gaussienne pour des modula-

tions non binaires. En�n, nous illustrons les gains en termes de taux d'erreurs binaires des codes ainsi

optimisés en comparaison avec des codes non optimisés.

• Chapitre 4: Dans le chapitre 4, nous étudions la seconde méthode permettant d'améliorer les débits,et ce en élargissant la bande du signal aux dépens d'une augmentation de la sélectivité en fréquence des

�ltres à bord du satellite. A�n de réduire les interférences issus de cette augmentation de bande, nous

présentons une forme d'onde permettant des traitements fréquentiels simpli�és. Cette forme d'onde a

les avantages d'une modulation mono-porteuse en termes de �uctuations d'enveloppe, et les avantages

d'une modulation multi-porteuses en termes d'égalisation fréquentielle simpli�ée. Dans le cadre de notre

étude, nous présentons un modèle fréquentiel et son équivalent en temporel pour cette forme d'onde, ce

qui permet de dériver des formules analytiques de densité spectrale de puissance. De plus, nous étudions

les interférences résiduelles après égalisation linéaire et dérivons des formules analytiques de rapport

signal à bruit plus interférences qui nous permettent de prédire les performances du système.

4 Introduction

Contributions principales

Les contributions principales de cette thèse sont résumées comme suit:

• Chapitre 1: Nous étudions l'impact des paramètres du système sur la représentation en série de Volterrade l'interférence non linéaire du canal satellite. Nous dérivons aussi un modèle, en nous basant sur des

paramètres souhaités du système, qui constituera notre canal de test plus tard dans le manuscrit.

• Chapitre 2: Nous dérivons des égaliseurs temporels et fréquentiels linéaires et non linéaires du canal deVolterra et comparons les complexités. Nous présentons de nouveaux résultats sur l'égalisation hybride

temps-fréquence appliquée au modèle de Volterra.

• Chapitre 3: Nous présentons l'égalisation itérative linéaire dans le domaine temporel [Benammar et al., 2013b]ainsi que dans le domaine fréquentiel [Benammar et al., 2014a]. Nous modélisons les sorties de l'égaliseur

par un mélange de Gaussiennes dont nous dérivons les paramètres pour de la détection sur un canal

Gaussien. Cette approximation est utilisée pour l'optimisation du code canal que nous appliquons à

di�érentes classes d'égaliseurs optimaux [Benammar et al., 2014b] et sous-optimaux.

• Chapitre 4: Nous dérivons un modèle temporel généralisé de la forme d'onde mono porteuses parmultiplexage à division orthogonale en fréquence (SC-OFDM). Nous proposons en outre des formules

analytiques de la densité spectrale de puissance [Benammar et al., 2013a] et du rapport signal à bruit

plus interférences pour ce type de forme d'ondes.

5

Introduction

For more than half a century, satellite systems have been conveying data over large and remote areas. Providing

high throughput satellite communications is a challenging aspect in the evolutions of next generation satellite

technologies be it for telecommunications, positioning or earth observation, .

When designing communication systems, many features interplay in dimensioning the link capacities. More

speci�cally, depending on the nature of the satellite service (�xed, mobile, broadcast), the transmission link

(up or down-link), the available bandwidth, and the acceptable complexity, solutions providing high spectral

and power e�cient satellite communications may di�er.

The power e�ciency measures the robustness to noise impairments and is usually related to the minimum

distance of the code and modulation. The spectral e�ciency characterises the communication throughput

per occupied bandwidth and thus is related to the cardinality of the modulation and the rate of channel

coding. There usually exists a trade-o� between achieving a good power and spectral e�ciency simultaneously.

However, emerging advances in iterative processing have allowed reducing this trade-o� by exploiting an

additional degree of freedom which is the system complexity. In this thesis, we position ourselves in the forward

link of a broadcast satellite system. For such a system, the key component is the satellite transponder which

comprises multiplexing �lters and the satellite ampli�er. Indeed, since there are constraints on the available

power and the maximum adjacent channel interference allowed, the transponder is usually the bottleneck in the

design of broadcasting satellite systems. We thus investigate the impact of using two methods for increasing

the throughput on the satellite transponder and thus on the overall system performance.

On the one hand, increasing the throughput by using high order modulations yields to higher signal �uctuations

which give rise to non linear interference when the signals are ampli�ed by a nearly saturated satellite ampli�er.

This kind of interference is also encountered in magnetic recording channels and �ber optical communications.

On the other hand, increasing the throughput can also be carried out by increasing the symbol rate. Thus,

the signal bandwidth may exceed the satellite multiplexing �lters bandwidths leading to increased frequency

selectivity in the satellite channel.

For both scenarios, an analysis of the nature and amount of interference is necessary in order to adopt adequate

processing. Mitigation of the satellite channel impairments can be carried out either at the transmitter through

pre-compensation and/or pre-distortion, or at the receiver by means of equalization. In this thesis, we are

interested in receiver design for non linear satellite channels and the advanced waveforms allowing for a more

e�cient equalization at the receiver.

6 Introduction

Structure of the manuscript

• Chapter 1: This chapter presents a description of the satellite systems under study. We are interestedin broadband satellite services and more speci�cally in the Digital Video and Broadcasting standards.

We present the DVB-S2 Amplitude and Phase Shift Keying schemes as well as related channel coding.

We then present the satellite transponder constituting components, and show the impact of some system

parameters such as the �lters roll-o�s, the signal bandwidth and the input back-o� on the behaviour of

the transponder. Moreover, we derive analytical symbol based description of the non linear interference

by means of in�nite Volterra series. For complexity considerations, this decomposition is truncated to

reasonable third and �ve orders and the impact of this truncation on the model accuracy is analysed.

Last but not least, we present the frequency domain equivalent Volterra series decomposition which is

used later in the manuscript for frequency domain based processing.

• Chapter 2: This chapter deals with non iterative equalization for the non linear Volterra satellite chan-nel. In the �rst part, we present a Markov chain description of the Volterra channel which then allows for

the derivation of optimal symbol and sequence equalization. Due to the exponential complexity of the

optimal equalizers, we investigate on sub-optimal linear and non linear equalization schemes in the time

domain. More speci�cally, we derive expressions for the linear Minimum Mean Square Error estimator

and two non linear Volterra decision feedback equalizers for the non linear channel. Then, similarly to

the time domain, we present frequency domain linear and non linear equalization schemes. We speci�-

cally derive novel results on the hybrid time and frequency domain equalizer. Then, we compare these

implementations in terms of bit error rates and complexity.

• Chapter 3: This chapter addresses iterative equalization and decoding for the non linear satellitechannel. In the �rst part, we remind results on optimal iterative equalization for channels described

by a trellis which we have shown to be the case of the Volterra non linear channel. Then, we derive

novel results on time domain linear iterative equalizers based on the Volterra channel, and investigate

on lower complexity approximations for the linear equalizer. We also analyse the frequency domain low

complexity iterative equalizers. In a second part, we design and optimize the channel code so as to �t

the equalizers output by the EXIT-chart curve �tting technique. To do so, we model the output of the

equalizer as a mixture of Gaussians which we show is more accurate than the Gaussian approximation.

Finally we illustrate the improvement in error rates for these approximations in comparison with non

7

optimized receivers.

• Chapter 4: In the chapter 4, we address the second throughput increasing technique relying on en-larging the signal bandwidth at the expense of increased frequency selectivity. Thus, to e�ciently cope

with the generated interference, we present a suitable single carrier transmission scheme which allows for

simple frequency domain equalization at the receiver. We present both a frequency domain and a novel

time domain model for this single carrier modulation which allows us to investigate on the spectrum

characteristics and derive analytical expressions for its spectral density. Moreover, we investigate on the

residual interference when linear equalizers are used and derive analytical expressions for the signal to

interference and noise ratio, which allows for a good prediction of the system performance.

Main contributions

The main contributions of this thesis can be summarised as follows:

• Chapter 1: We study of the impact of the system parameters on the representation of the non linearsatellite channel by Volterra series, and derive a test channel model given some system parameters which

will be used later in simulations.

• Chapter 2: We derive a novel hybrid time and frequency domain equalizer for the Volterra descriptionof the non linear interference and a detailed complexity analysis for di�erent classes of equalizers.

• Chapter 3: We investigate on iterative linear time domain MMSE equalization [Benammar et al., 2013b]and frequency domain equalizers for the non linear satellite channel [Benammar et al., 2014a].

We also model the output of the equalizer as a Gaussian Mixture instead. This approximation is used

in the code design and optimization for a class of optimal and sub-optimal iterative receivers for the non

linear channel [Benammar et al., 2014b].

• Chapter 4: We derive a time general representation of the Single Carrier -Orthogonal Frequency DivisionMultiplexing (SC-OFDM) modulation. We also derive analytical expressions for the spectral density

[Benammar et al., 2013a] and signal to interference noise ratio for SC-OFDM.

8 Introduction

Chapter 1

Digital communications over satellite

channels

Contents1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Introduction (french) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Satellite communication system . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Broadcasting Satellites standards . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 DVB-S standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.2 DVB-S2 standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.3 DVB-S2X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Transponder modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5.1 TWT and SSP Ampli�ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5.2 Input and output multiplexers . . . . . . . . . . . . . . . . . . . . . . . . . . 201.5.3 Saturation levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6 Impact of system parameters on the non linear channel . . . . . . . . . . 22

1.6.1 Impact of IBO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.6.2 Impact of the root raised cosine roll-o� . . . . . . . . . . . . . . . . . . . . . 231.6.3 Impact of the signal bandwidth in the presence of IMUX and OMUX . . . . 24

1.7 Non linear satellite channel models . . . . . . . . . . . . . . . . . . . . . . . 25

1.7.1 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.7.2 Volterra model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.7.3 Volterra coe�cients design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.7.4 Volterra decomposition for test channels . . . . . . . . . . . . . . . . . . . . . 33

1.8 Frequency domain Volterra model . . . . . . . . . . . . . . . . . . . . . . . 34

1.8.1 Multi-dimensional Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . 351.8.2 Frequency domain Volterra model . . . . . . . . . . . . . . . . . . . . . . . . 36

1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.10 Conclusion (french) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9

10 Chapter 1 - Digital communications over satellite channels

1.1 Introduction

New generation satellite broadcasting trends recommend using high order modulations to increase the spectral

e�ciency of the satellite link. Such high order modulations can be designed to allow a better resistance to

noise (compared to amplitude shift keying), such as the new modulations suggested in the second generation

broadcasting system. However, increasing the power e�ciency does not come at no cost, since the resulting new

waveforms have higher signal dynamics. This increased �uctuation has a direct impact on the operating region

of non linear ampli�ers on board satellites. Indeed, if the signal has large �uctuations, a non negligible back-o�

is required to the saturation power of the ampli�er. The introduced back-o�s decrease the energy e�ciency

of the power ampli�er and thus impacts the overall link budget. If instead small back-o�s are considered

despite the large signal dynamics, the ampli�ed signal is distorted and unless suitable mitigation techniques

are used, the link budget is again impacted. It is thus important to understand the di�erent technical issues

involving the use of high order modulations and the corresponding RF design challenges. To do so, we start

by introducing the context of satellite broadcasting systems. We then study the impact of di�erent system

con�gurations on the distortions induced by the non linear power ampli�er. Then, we develop an analytical

symbol based model for these distortions. Finally we investigate on equivalent representations of the non linear

channel in the frequency domain, which will help us later develop frequency domain mitigation techniques.

1.2 Introduction (french)

Les nouvelles tendances concernant la di�usion par satellites, suggèrent l'utilisation de schémas de modulations

d'ordre élevé a�n d'améliorer l'e�cacité spectrale des communications par satellites. Ces modulations d'ordre

élevé peuvent être conçues a�n de permettre une meilleure résistance au bruit (comparées aux modulations

d'amplitude) à l'instar des modulations proposées dans la seconde génération des systèmes de di�usion. Cepen-

dant, l'amélioration de l'e�cacité en puissance n'est pas gratuite, puisque les nouvelles formes d'ondes ainsi

générées ont une plus forte dynamique du signal. Cette �uctuation additionnelle a un impact direct sur la zone

de fonctionnement des ampli�cateurs non linéaires présents à bord des satellites. En e�et, si le signal a une

forte �uctuation, un recul non négligeable vis à vis de la puissance de saturation est nécessaire. Ceci représente

un inconvénient majeur, puisque ces reculs diminuent l'e�cacité énergétique des ampli�cateurs, et in�uent sur

le bilan de liaison global. Si par contre, de faibles reculs par rapport à la puissance de saturation sont utilisés,

le signal ampli�é est distordu et à défaut de techniques adéquates d'élimination de ces distorsions, le bilan de

liaison est impacté. A�n d'illustrer les enjeux liés à ces nouvelles modulations, nous commencerons par une

1.3 - Satellite communication system 11

introduction des systèmes de di�usion par satellite. Nous étudierons ensuite l'e�et de certains paramètres sur

les distorsions générées par le canal nonlinéaire. Nous présenterons ensuite un modèle analytique au rythme

symbole pour ces distorsions. Finalement, nous éudierons une représentation équivalente de ce modèle dans le

domaine fréquentiel, ce qui nous permettra plus tard de développer des méthodes de traitement des distorsions

dans le domaine fréquentiel.

1.3 Satellite communication system

A classical transparent satellite communication system consists of three distinct blocks:

• The ground station which is usually referred to as the hub.It consists for uplink communications of a transmitting dish or antenna fed by signals aggregated from

di�erent baseband signals. A power ampli�er is used to cope with the sharp attenuation incurred by

the transmitted signal when propagating through long distances to the satellite.

• The satellite or the space segment.It consists of three di�erent components namely, the fuel component responsible for the propulsion, the

Telemetric Tracking and Command (TT & C) system and the communication payload which usually

contains several transponders. The TT & C system is used for all operations and commands concerning

the deployment and the maintenance of the satellite in orbit. The transponder generally designated by

the payload is responsible for all communications with the outside environment and thus for the Radio

Frequency (RF) communications.

Satellites can be classi�ed into Fixed Satellite Services (FSS), Mobile Satellite Services (MSS) and

Broadband Satellite Services (BSS). In this thesis, we focus on BSS systems with transparent satellites

which unlike regenerative satellites do not demodulate the baseband signal but only repeat the incoming

RF signal to the appropriate downlink channel with power ampli�cation.

• The receiving station which can be a ground station, individual antennas or terminals directly locatedat the customer. For television applications, the satellite broadcasts signals over a wide area which can

then be received by a large number of users with the use of small receiving antennas.

For satellite communications, the satellite transponder is generally the bottleneck of the system design, since

it has limited resources and thus any required base-band processing should be located either at the transmitter

station or at the receiving station.

12 Chapter 1 - Digital communications over satellite channels

Figure 1.1: DVB-S functional block diagram

1.4 Broadcasting Satellites standards

Broadcasting satellites are usually located in the geostationary orbit at about 37.000 km. The European

Telecommunications Standards Institute (ETSI) committee has issued many standards regulating satellite

broadcasting, depending on the type of the conveyed data.

Among the technologies of satellite broadcasting standards, we are interested in this thesis in Digital Video

Broadcasting (DVB) standards and more speci�cally in DVB-S its 2nd generation evolution DVB-S2.

1.4.1 DVB-S standard

The standardised DVB-S [EN, 1997] is illustrated in Figure 1.1. It o�ers a 36MHz communication channel

bandwidth and uses power e�cient modulations namely the Binary Phase Shift Keying (BPSK) and Quadra-

ture Phase Shift Keying (QPSK). Concatenated error correcting codes are used for channel coding. The

inner code is based on a set of punctured Convolutional Codes (CC) constructed from a 1/2-rate convolu-

tional code with constraint length equal to 7. The outer code is a shortened (N,K) Reed Solomon (RS) code

(N = 204,K = 188, T = 8) constructed on the Galois Field GF (28) from a RS code (N = 255,K = 239, T = 8)

where is the length of the codewords, and K is the length of the information symbols and T is the correction

capacity. It should be noted that the RS code shortening is realised by appending 51 null bytes to each block

of 188 bytes and discarded at the end of the coding/decoding process. Since the RS code is systematic, the null

bytes can be easily inserted and discarded at both ends of the coder and decoder. A convolutional interleaver

is inserted between the two channel codes to o�er a better correction capacity to the overall concatenated

channel code.

1.4.2 DVB-S2 standard

The DVB-S2 standard depicted in Figure 1.2 has been proposed as a spectrally and power e�cient transmission

technology through using Amplitude and Phase Shift Keying (APSK) modulations and a class of capacity

approaching block codes: Low Density Parity Check (LDPC) codes. The achieved system capacity gain over

1.4 - Broadcasting Satellites standards 13

Figure 1.2: DVB-S2 functional block diagram

the �rst generation DVB-S can reach 30%. Additionally, compared to the DVB-S standard, DVB-S2 o�ers

the possibility of adapting the modulation and coding formats to the link quality with the so-called Adaptive

Coding and Modulation (ACM) functionality.

DVB-S2 modulation schemes

The APSK modulation has been introduced in the DVB-S2 standard for its good trade-o� between spectral and

power e�ciency. Indeed, for an equivalent spectral e�ciency η, the APSK modulation has better robustness to

Gaussian noise compared to PSK modulation. This power e�ciency gain is achieved through dispatching the

symbols over multiple rings allowing a better separation distance between symbols and yet carrying equivalent

number of bits per modulated symbol. However, when compared to Quadrature an Amplitude modulations

(QAM) constellations, APSK is less noise resistant but o�ers lower signal �uctuations which is a valuable

feature especially for power ampli�ers. Since APSK modulation symbols are distributed over multiple rings,

it is convenient to de�ne the ratio γ which characterises the relation between modulations radii. As such, for

a 16APSK which consists of two concentric rings, the ratio γ writes as:

γ =R2R1

(1.1)

where R2 and R1 are the outer and inner rings radii respectively. For a 32APSK, the DVB-S2 standard de�nes

two ratios namely γ1 and γ2 describing the pairwise ratios and writing as follows:

γ1 =R2R1

and γ2 =R3R2

(1.2)

where R3, R2 and R1 are the outer, intermediate and inner rings radii respectively. An illustration of the

di�erent proposed modulation schemes can be found in Figure 1.3. The modulation symbols si for these

schemes can be written in a generic form as follows, :

si = Rn exp(j2πφk)fori = 1 : M (1.3)

where Rn is one of the radii of the modulation scheme and φk is one of the allowed modulation phases for the

ring radius Rn. Table 1.1 presents the di�erent radii and phases for DVB-S2 mapping sets. The mapping used

14 Chapter 1 - Digital communications over satellite channels

−1 0 1−1

−0.5

0

0.5

1Q

uadr

atur

e

QPSK

−0.5 0 0.5

−0.5

0

0.5

Qua

drat

ure

8PSK

−2 0 2

−2

−1

0

1

2

Qua

drat

ure

In−Phase

16 APSK

−5 0 5

−5

0

5

Qua

drat

ure

In−Phase

32 APSK

Figure 1.3: Scatter plot of the DVB-S2 modulation γ = 2.85 and γ1 = 2.84 and γ2 = 5.27

for PSK modulation is a Gray mapping whereas the mapping used for APSK modulations is a quasi-Gray

mapping. It is important to notice that the performance of the APSK modulation, depends on the ratios

γi, the number of constellation symbols on each ring, and the phase o�sets between symbols. The minimum

distance which measures the robustness to noise can be optimized to yield the targeted performance.

DVB-S2 coding schemes

The inner code of the DVB-S2 channel consists of a class of capacity approaching codes namely the LDPC

codes. The proposed code is systematic with KLDPC input bits and NLDPC coded bits. The standard

suggests two frame lengths consisting of short frames of length NLDPC = 16200 and long frames of length

NLDPC = 64800.

The outer code is a Bose, Ray-Chaudhuri et Hocquenghem (BCH) block code with parameters (NBCH ,KBCH , T )

where T is the correction capacity of the code. A block interleaver is used between the two channel codes to

cope with burst errors. This interleaver writes input stream in a matrix column-wise, and reads the elements

line-wise. The dimensions of the interleaver matrix are given in [EN, 2009] for normal and short frames.

1.4 - Broadcasting Satellites standards 15

Radii Phases Mapping in decimal

QPSK R = 1 k π2 +π4 [0, 2, 3, 1]

8PSK R = 1 k π4 +π4 [0, 4, 6, 2, 3, 7, 5, 1]

16APSK R1 = 1 kπ2 +

π4 [12, 14, 15, 13]

R2 = γ kπ6 +

π12 [4, 0, 8, 10, 2, 6, 7, 3, 11, 9, 1, 5]

32APSK R1 = 1 kπ2 +

π4 [17, 21, 23, 19]

R2 = γ1 kπ6 +

π12 [16, 0, 1, 5, 4, 20, 22, 6, 7, 3, 2, 18]

R3 = γ1γ2 kπ8 [24, 8, 25, 9, 13, 29, 12, 28, 30, 14, 31, 15, 11, 27, 10, 26]

Table 1.1: DVB-S2 modulation schemes parameters for QPSK, 8PSK, 16APSK and 32ASPK

Code rate Modulation coding/spectral e�ciency γ

2/3 2, 66 3, 15

3/4 2, 99 2, 85

4/5 3, 19 2, 75

5/6 3, 32 2, 70

8/9 3, 55 2, 60

9/10 3, 59 2, 57

Table 1.2: Optimum constellation radius ratio for AWGN channel with 16APSK

In [De Gaudenzi et al., 2006], the overall DVB-S2 system has been optimized to yield the best system ca-

pacity. To do so, the radii of APSK modulations was jointly optimized with the coding rate to yield the

best spectral e�ciency. Table 1.2 and Table 1.3 summarize the designed optimal ratios γ and (γ1, γ2) for the

couples (coding rate, spectral e�ciency).

1.4.3 DVB-S2X

DVB-S2X [DVB-S2X, 2014] is an evolution of the standard DVB-S2 which relies on the same physical layer

characteristics regarding the types of modulations and channel codes. However, there are some di�erences in

the system parameters which can be summarised as follows:

• Small roll-o�s (0.05 and 0.1) can be used leading to up to 15% gain in the system throughput.

• Finer modulations and coding rates

• The modulation ring ratios can be jointly chosen with coding rates for given ampli�er back-o�s.

16 Chapter 1 - Digital communications over satellite channels

Code rate Modulation coding/spectral e�ciency γ1 γ2

3/4 3, 74 2, 84 5, 27

4/5 3, 99 2, 72 4, 87

5/6 4, 15 2, 64 4, 64

8/9 4, 43 2, 54 4, 33

9/10 4, 49 2, 53 4, 30

Table 1.3: Optimum constellation radius ratio for AWGN channel with 32APSK

Using amplitude and phase shift keying modulations as originally proposed in the DVB-S2 standard has

given rise to some challenges related to power ampli�ers e�ciency. Power ampli�ers are located both at

the transmitter (ground station) and on-board satellites. Yet, since there are less restrictions on the power

supply of a hub or a gateway on a forward link, the limiting power ampli�er is the one located in the satellite

transponder. For a better understanding of the ampli�ers e�ects, we present in the next section the satellite

transponder constituent elements and how they impact the channel non linearity.

1.5 Transponder modelling

In this section we are interested in the elements constituting a transponder, and more speci�cally, the power

ampli�er and the input and output multiplexers. The considered ampli�er is a memory-less device with a

frequency independent ampli�cation model. The input and output multiplexers placed before and after the

power ampli�er are meant to reject undesired spectral components. In the following, we give more insights on

the classes of ampli�ers and the multiplexing �lter responses.

1.5.1 TWT and SSP Ampli�ers

TWTA

Travelling Wave Tube Ampli�er (TWTA) are wideband microwave ampli�ers capable of amplifying a wide

range of frequencies. Figure 1.4 depicts the structure of a TWT ampli�er.

A cathode heated at thousands of degrees generates an electron beam which is accelerated by the anode using

a high potential. These electrons propagate into a vacuum cavity containing a helix related to RF inputs and

outputs. The interaction between the RF signal and the electron beam leads to a deceleration of the electrons,

whose kinetic energy is transferred to the RF signal which is then ampli�ed. At the end of their race, electrons

1.5 - Transponder modelling 17

Figure 1.4: Structure of a Travelling Wave Tube Ampli�er

are captured by the collector which receives the remaining electrons energy. TWTA have interesting high gain

and low noise characteristics which makes them suitable for RF ampli�cation [Gilmour, 2011].

The ampli�cation process of a TWTA is usually described using two functions: Amplitude to Amplitude (AM-

AM) and Amplitude to Phase (AM-PM). Theses functions relate the input amplitude to the output amplitude

and phase rotation respectively. To describe a TWTA, Saleh in [Saleh, 1981] presented a frequency-independent

model to characterise the AM-AM and AM-PM functions of a TWTA. The derived frequency-independent

model of the ampli�er only depends on the instantaneous input amplitude r. The AM-AM and AM-PM

functions write as follows [Saleh, 1981]:

AM −AM(r) = αar1 + βar2

AM − PM(r) = αφr.2

1 + βφr2(1.4)

where r is the input signal amplitude and αa, βa, αφ and βφ are design parameters which characterise the

AM-AM and AM-PM functions respectively. For instance, Figure 1.5 plots a TWTA ampli�er functions using

parameters αa = 1.9638, βa = 0.9945, αφ = 2.5293 and βφ = 2.8168. IBO and OBO stand for backo�s and

are presented in the next section.

18 Chapter 1 - Digital communications over satellite channels

−20 −15 −10 −5 0 5 10−15

−10

−5

0

Input Normalised Power [dB]

Out

put P

ower

[dB

]IBO

OBO

Figure 1.5: AM-AM characteristic using Saleh's model with parameters αa = 1.9638 and βa = 0.9945

SSPA

A Solid State Power Ampli�er (SSPA) is a device that uses Field E�ect Transistors (FET) and thus relies on

solid components unlike a TWTA which uses a vacuum tube. A SSPA is composed of serial/parallel combi-

nations of FETs which used alone would have delivered limited gain. It usually consists of four stages using

power combiners, dividers and medium power ampli�ers. The power combiners are dissipative components

which leads to a lower energy e�ciency as their number increases.

In a similar fashion to TWTAs, SSPAs can be described by two functions namely AM-AM and AM-PM. An-

alytical models which can be used for representing these functions are di�erent from Saleh's equations (1.4).

In fact, two analytical models describing the SSPAs nonlinearity can be found in literature, the Rapp model

[Rapp, 1991] [Costa and Pupolin, 2002] and the Ghorbani model [Ghorbani and Sheikhan, 1991] respectively.

On the one hand, the Rapp's model de�nes AM-AM and AM-PM equation as follows:

AM −AM(r) = v r(1 +

(vrA0

)2p) 22p p > 0, A0 ≥ 0, v ≥ 0

AM − PM(r) = αΦ(vr

A0

)4(1.5)

where r is the input signal amplitude, v is called the small-signal gain, A0 is the saturation amplitude level

and p is a factor that controls the smoothness of the curve before saturation. In the AM-PM conversion, αΦ is

generally set to zero since the phase rotation is considered negligible for a SSPA compared to TWTA. On the

1.5 - Transponder modelling 19

−20 −15 −10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Input normalised power [dB]

Pha

se [r

ad]

IBO

∆ φ

Figure 1.6: AM-PM characteristic using Saleh's model with parameters αφ = 2.5293 and βφ = 2.8168

other hand, Ghorbani's model presents a di�erent approximation of SSPA's AM-AM and AM-PM conversions.

These conversions write as:

AM −AM(r) = x1rx2

1 + x3rx2+ x4r

AM − PM(r) = y1ry2

1 + y3ry2+ y4r (1.6)

where r is the input signal amplitude, x1, x2, x3, x4 and y1, y2, y3, y4 are adjusting coe�cients. It can be

seen, that the Ghorbani model can be written as a Saleh decomposition for special values of the adjusting

parameters.

Comparison between TWTA and SSPA

For a comparison between the two classes of ampli�ers to be carried out, one needs to determine what is

the intended system use of these ampli�ers. Indeed, each of the TWTAs and SSPAs have advantages and

drawbacks, which implies they can only be assessed with regard to the application requirements. To do

so, authors in [Maral and Bousquet, 2002] present a set of comparative characteristics between TWTAs and

SSPAs which are presented in Table 1.4. On the one hand, SSPAs are lighter than TWTA but have less energy

e�ciency which makes them more suitable to small satellites in low orbits such as observation satellites. On

the other hand, TWTAs have better e�ciency and can operate over a wide range of frequencies which makes

them suitable for telecommunication applications such as broadcasting satellites. In the remaining of this

20 Chapter 1 - Digital communications over satellite channels

Characteristic TWTA SSPA

Operating band (GHz) C, Ku, Ka L,C

Saturated power output (W) 20− 250 20− 40Gain at saturation (dB) ∼ 55 70− 90Intermodulation product relative level (C/N)IM3 (dB) 10− 12 14− 18AM/PM conversion coe�cient Kp(/dB) 4.5 2

DC to RF e�ciency 50− 65 30− 45Mass including (kg) 1.5− 2.2 0.8− 1.5

Table 1.4: Comparative characteristics for TWTA and SSPA

−40 −30 −20 −10 0 10 20 30 40−60

−50

−40

−30

−20

−10

0

Frequency [Mhz]

Gai

n [d

B]

IMUXOMUX

Figure 1.7: Gain for IMUX and OMUX

thesis, the considered ampli�ers will be thus TWTAs, which will be modelled by Saleh's representation as in

(1.4).

1.5.2 Input and output multiplexers

The Input MUltipleXer (IMUX) is a band-pass �lter which aims at removing adjacent channels interference

caused by other input channels. The Output MUltipleXer (OMUX) consists of a band-pass �lter which rejects

the out-of-band radiation due to the spectral growth induced by the non linear ampli�er processing. The

reader is referred to Section 1.6.1 for more details about the impact of the ampli�cation on the power spectral

density. Figure 1.7 and Figure 1.8 illustrate the gain and group delay of typical 36MHz IMUX and OMUX

1.6 - Impact of system parameters on the non linear channel 21

−40 −30 −20 −10 0 10 20 30 40−5

0

5

10

15

20

25

30

35

40

Frequency [Mhz]

Gro

up d

elay

[ns]

IMUXOMUX

Figure 1.8: Group delay for IMUX and OMUX

�lters. The phase response is applied by integrating the group delay (GD) following:

φ(ω) =

∫GD(ω)dω (1.7)

where ω designates the frequency.

1.5.3 Saturation levels

In order to de�ne the operating point of a power ampli�er, it is useful to introduce two metrics which char-

acterise the back-o� towards the input saturation power and the output power respectively. We thus de�ne

the Input Back-O� (IBO) as the ratio between the input power Pin and the input saturation power Pin,sat as

follows:

IBO = −10 log10Pin

Pin,sat(1.8)

Similarly we de�ne the Output Back-O� (OBO) as the ratio between the output power Pout and the maximum

output power delivered by the PA as follows:

OBO = −10 log10Pout

Pout,sat(1.9)

Figure 1.5 and Figure 1.6 plot the TWTA transfer characteristics with its IBO and OBO operating points.

22 Chapter 1 - Digital communications over satellite channels

Figure 1.9: Satellite channel modelling

1.6 Impact of system parameters on the non linear channel

Let us consider the system depicted in Figure 1.9. A stream of symbols xn with a symbol duration Ts is pulse

shaped by a root raised cosine �lter with roll-o� α writing as:

h(t) =2α

π√Ts

cos[(1 + α)π tTs

]+

sin[(1−α)π tTs ]4α tTs

1−(

4α tTs

)2 (1.10)The resulting transmit signal is then sent to a satellite transponder where it is �rst �ltered with the IMUX,

ampli�ed by the HPA and then �ltered out by the OMUX. The output signal y(t) is broadcast to the receiving

station where an additive white Gaussian noise with variance σ2w. A matched receive �lter is used before

sampling at the corresponding Nyquist timing t0 + nTs. The resulting symbols zn depend on the system

parameters such as the root raised cosine roll-o�, the signalling rate and on the HPA back-o�, which hence

has an impact on the resulting system performance. It is thus interesting to investigate on the impact of each

of the aforementioned system parameters on the received symbols. To do so, we set the noise variance to zero

to assess only the non linear interference.

1.6.1 Impact of IBO

The IBO as de�ned in Section 1.5.3 is a key factor in the amount of ISI generated by the non linear satellite

channel. Figure 1.10 plots di�erent scatter-plots for a 16APSK using a root raised cosine of roll-o� α = 0.2

and for IBO = 0, 3, 6, 9dB. The �gure shows that the amount of ISI decreases with an increasing IBO which

is due to the linear-like behaviour of the HPA for very high back-o�s. It should be noted that the nonlinear

ISI is not isotropic, since there seems to be a correlation between real and imaginary parts of the ISI owing to

its elliptical shape. To illustrate this property, Figure 1.11 and Figure 1.12 plot the probability distribution

function of the power of ISI around the average constellation point, which is often called a centroid. It can

be seen that for low IBO values, the distribution of ISI of outer rings symbols is not a circular Gaussian

1.6 - Impact of system parameters on the non linear channel 23

−2 −1 0 1 2−2

−1

0

1

2

In−Phase

Qua

drat

ure

IBO = 0dB

−2 −1 0 1 2−2

−1

0

1

2

Qua

drat

ure

In−Phase

IBO = 3dB

−2 −1 0 1 2−2

−1

0

1

2

In−Phase

Qua

drat

ure

IBO = 6dB

−2 −1 0 1 2−2

−1

0

1

2

In−Phase

Qua

drat

ure

IBO = 9dB

Figure 1.10: Constellation for 16APSK with roll-o� α = 0.2

distribution, since its imaginary and real part histograms are not similar. Yet, with increasing IBO values, the

distribution of ISI becomes circular Gaussian which again is due to the linear operating region of the HPA for

high IBOs.

When being ampli�ed, the power spectral density of the signal is also modi�ed by the non linear processing.

The PSD is expanded leading to novel frequency components which is generally referred to as "spectral growth".

The amount of spectral growth depends on the IBO values and is depicted in Figure 1.13. It is clear that the

smaller the IBO, the larger the spectral growth, due to the increased nonlinearity of the power ampli�er.

1.6.2 Impact of the root raised cosine roll-o�

The root-raised cosine �lter introduces a memory in the satellite system which combined with the memoryless

non linear ampli�er leads to the observed ISI in Figure 1.10. One can wonder what is the impact of the roll-o�

24 Chapter 1 - Digital communications over satellite channels

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x

PD

F(x

)

Real part histogramReal part theoryImag part histogramImag part theory

Figure 1.11: PDF of real and imaginary parts of one outer ring centroid at IBO = 0dB

Symbol rate \ IBO 0dB -3dB -6dB -9dB27.5 Mbauds 0.0561 0.0440 0.0373 0.0326

30 Mbauds 0.0863 0.0695 0.0585 0.0552

32.5 Mbauds 0.1505 0.1282 0.1186 0.1081

Table 1.5: ISI variance for di�erent signal bandwidths

of the root raised cosine on the ISI power and shape. Figure 1.14 depicts the ISI variance for one of the outer

ring constellation symbols using di�erent IBOs and roll-o�s. The higher the roll-o�s, the lower the side lobes

of the impulse response are, and thus the less the impact of the non linearity. It can be further noticed, that

the dependence of the non linear interference power on the roll-o� decreases for large IBO values. Ultimately,

the roll-o� in�uence vanishes for very high IBO values, since the ampli�er consists then of a constant gain.

Thus, this would yield to interference-free received symbols, since the root-raised cosine �lters used in the

chain are Nyquist shapes.

1.6.3 Impact of the signal bandwidth in the presence of IMUX and OMUX

The IMUX and OMUX �lters present on both sides of the satellite non linear ampli�ers control the input and

output bandwidth of the satellite transponder signals. The multiplexing �lters de�ned in Section 1.5.2 were

designed for a 33MHz signal bandwidth, which for a certain roll-o� α leads to a desired symbol rate Rs =

33/(1 + α)Mbauds. For example with a roll-o� factor α = 0.2, the desired symbol rate is Rs = 27.5Mbauds.

1.7 - Non linear satellite channel models 25

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20

1

2

3

4

5

6

7

8

x

PD

F(x

)

Real part histogramReal part theoryImag part histogramImag part theory

Figure 1.12: PDF of real and imaginary part of one outer ring centroid at IBO = 9dB

If the symbol rate is higher than 27.5Mbauds, the �lters will be more frequency selective and thus will lead to

increased interference. In Table 1.5, the amount of ISI is expressed by means of the MSE between the received

symbols and their centroids and di�erent signal bandwidths are investigated. It can be seen that the larger

the symbol rate in comparison with the reference Rs = 27.5Mbauds, the higher the MSE.

1.7 Non linear satellite channel models

The issue of modelling the satellite transponder e�ects on the received demodulated signals has been investi-

gated in many previous studies. Indeed, modelling the satellite non linear channel is of primary importance in

order to adopt the adequate processing techniques in order to cope with adverse e�ects occurring in the satel-

lite link. Dependin